A tournament on a set X is an irreflexive binary relation RÌX×X such that, for every x¹y in X, exactly one of
R(x,y) and R(y,x) holds. A pseudofinite field F interprets a
tournament by the formula $z : (x-y)=z2. The automorphism
group of any field interpreting a 0-definable tournament can not
have any involutions.

To generalize this observation, we will examine the effects of
interpreting such structures on the automorphism groups of certain
pseudofinite fields.

We consider to what extent the role played by weakly o-minimal
theories among all densely ordered theories is analogous to the role
played by o-minimal theories among the definably complete densely
ordered theories. In particular we ask whether certain results
indicating that any sufficiently well-behaved definably complete
theory is o-minimal or close to o-minimal may be generalized by
dropping the assumption of definable completeness and weakening the
conclusion from "o-minimal or close to o-minimal" to "weakly
o-minimal or close to weakly o-minimal". We show through example
that this is not the case.

Let G be a semiabelian variety defined over a number field K. Let
X be a subvariety of G defined over Kalg. The
Manin-Mumford Theorem describes the intersection of X(Kalg)
with the torsion subgroup Gtor of G. More precisely, if X
is an irreducible subvariety and X(Kalg) ÇGtor is
Zariski dense in X, then X is a translate of an algebraic subgroup
of G by a torsion point. In the present talk we show that we obtain
the same conclusion about X assuming only that it contains a Zariski
dense set of points of small height. Because all torsion points of
G have height 0, we obtain that Manin-Mumford Theorem is a
particular case of the result we present. Finally, we present a
positive characteristic version of the Manin-Mumford Theorem in the
context of Drinfeld modules.

An analogue for valued fields of Hilbert's Seventeenth Problem asks
for an algebraic characterisation of rational functions which map a
definable subset of the field into the valuation ring. In this talk,
I will describe a model-theoretic solution to this problem which is
uniform across different theories of valued fields. I will apply it
to algebraically closed valued fields, real closed valued fields, and
model complete theories of difference and differential fields with a
valuation.

We show that the germ of a Riemann map (i.e., a biholomorphic
map from a simply connected domain in the complex plane onto the unit
ball) at an analytic corner of angle greater than 0 can be realized in
a certain quasianalytic class, used by Ilyashenko in his solution of
Hilbert's 16th problem. With this we are able to show that the
Riemann map from a simply connected domain which is semianalytic and
bounded, is definable in an o-minimal structure under some condition
on the singularities of the domain.

An integer part (IP for short) Z of an ordered field K is
a discretely ordered subring, with 1 as the least positive element,
and such that for every xÎK, there is a zÎZ such that z£x < z+1. Mourgues and Ressayre establish the existence of an IP for
any real closed field K by showing that there is an order preserving
embedding j of K into the field of generalized power series
k((G)) such that j(K) is a truncation closed subfield (here
k is the residue field and G the value group of K). An IP of
K obtained in this way (i.e., from a truncation closed
embedding) is called a truncation integer part of K. IPs
appear naturally in model theoretic arithmetic, algebra and analysis;
e.g. Shepherdson showed that IPs of real closed fields are
precisely the models of a fragment of Peano Arithmetic called Open
Induction, whereas truncation IPs played a crucial role in Ressayre's
investigations of the model theory of the real exponential field. In
this talk, we analyze IPs from a valuation theoretic viewpoint and
summarize their main special features. We investigate their
connection to special (additive) complements of valuation rings of
ordered fields. This approach reveals new interesting valuation
theoretic properties of arbitrary valued fields (not just
ordered fields); depending on whether such special complements exist.
We discuss these properties and their implications, thereby giving an
intrinsic valuation theoretic interpretation of truncation closed
embeddings in fields of power series.

There exist closed EÍR such that (R,+,·,E) defines a Borel isomorph of (R,+,·,N), and so defines sets of every projective level, yet does
not define N, even when (R,+,·,E) is further
expanded by all subsets of every cartesian power of E.

In this paper we explore a nonstandard formulation of Hausdorff
dimension. By considering an adapted form of the counting measure
formulation of Lebesgue measure, we prove a nonstandard version of
Frostman's lemma and find that Hausdorff dimension can be computed
through a counting argument rather than by taking the infimum of a
sum of certain covers. This formulation is then applied to obtain a
simple proof of the doubling of the dimension of certain sets under
a Brownian motion. In addition, the fractal properties of the rapid
points of Brownian motion are explored using the new method,
strengthening a result of Orey and Taylor's.

Let j: x®j(x), x > a be a solution at
infinity of an algebraic differential equation of order n,
P(x,y,y¢,...,y(n)) = 0. We establish a geometric criterion so
that the germ at infinity of j, together with that of the
identity function on R, belongs to a common Hardy field.

More precisely, under the hypothesis that ¶P/¶y(n)( x,j(x),j¢(x),...,j(n)(x) ) is
never zero, the criterion is the following non-oscillating property:
for any polynomial QÎR [x,y,y¢,...,y(n-2)], the
function x®Q ( x,j(x),j¢(x),...,j(n-2)(x) ) has a definite sign for x >> 0. Immediate
applications for differential equations of order one or two are given.

A result of Johnson, Reinhart and Rubel (1995) shows that, unlike the
one variable case, it is not possible in general to approximate
solutions to partial differential equations via finite transcendence
extensions. We indicate the relationship of this phenomenon to types
in the model theory of partial differential fields in characteristic
zero.